Estimate Propensity Scores for Exposure Assignment

Why Model Exposure Assignment?

In an observational study, people with different measured characteristics may have different probabilities of receiving each exposure. For example, age or baseline disease severity may be related both to treatment choice and to a later outcome. A propensity score summarizes, for a person's measured covariate profile, the probability of the exposure they receive—or, in a multi-level setting, the probability of each exposure level. It describes the assignment process; it is not an outcome prediction or an estimate of a treatment effect.

The focus here is on estimating those probabilities. A propensity score model is built using exposure assignment as its response and measured, pre-exposure covariates as predictors. The causal question and target population should already be clearly defined, and the chosen covariates should be guided by the causal structure of the study. A model that predicts exposure well is not automatically a model that removes confounding.

Definition: Let \(A\) denote exposure and \(X\) the measured covariates recorded before exposure. The propensity score for exposure level \(a\) is \(e_a(X)=P(A=a\mid X)\). For a binary exposure coded \(A=1\) or \(A=0\), the usual propensity score is \(e(X)=P(A=1\mid X)\); the probability of the other level is \(1-e(X)\). An estimated propensity score, written \(\hat e_a(X)\), is a fitted estimate of this conditional probability from the observed data.

Binary Exposure: Logistic Regression

For a binary exposure, a common assignment model is logistic regression. Let \(X_1,\ldots,X_p\) be measured pre-exposure covariates, and suppose the model specifies

\log\left(\frac{P(A=1\mid X)}{1-P(A=1\mid X)}\right) =\beta_0+\beta_1X_1+\cdots+\beta_pX_p.

The left side is the log odds of receiving exposure level 1. If the fitted linear predictor for a covariate profile is \(\hat\eta(X)\), converting the fitted log odds to a probability gives

\hat e(X)=\frac{\exp\{\hat\eta(X)\}}{1+\exp\{\hat\eta(X)\}}.

This model assumes that the specified log odds are adequately represented by the chosen covariates and their functional forms. For example, including age only as a linear term assumes a linear relationship between age and log odds, not necessarily between age and probability. The exposure categories must be coded correctly, the predictors must be available before exposure, and the data structure must be handled appropriately—for example, dependence from clustering should not be ignored when it matters. Positivity is a separate causal-design condition: within covariate profiles relevant to the target population, the exposure levels being compared must have nonzero probabilities. A fitted regression does not guarantee positivity.

Worked Example: A Binary Exposure Probability

Suppose a hypothetical clinic dataset records whether each person received a new treatment (\(A=1\)) or usual care (\(A=0\)). A fitted logistic assignment model uses baseline age in decades, centered at 5 decades, and a baseline severity indicator \(S\) (1 for severe, 0 otherwise):

\hat\eta=-1.20+0.35(\text{age in decades}-5)+0.80S.

For a hypothetical 60-year-old person with severe baseline symptoms, age in decades is 6 and \(S=1\). The fitted log odds and propensity score are

\hat\eta=-1.20+0.35(6-5)+0.80(1)=-0.05, \qquad \hat e(X)=\frac{\exp(-0.05)}{1+\exp(-0.05)} =\frac{0.9512}{1+0.9512}\approx0.4875.

Under this fitted model, the estimated probability that a person with this measured profile received the new treatment is about 0.488, or 48.8%. The estimated probability of usual care is \(1-0.4875=0.5125\), or 51.3%. These are model-based assignment probabilities, not a claim that treatment caused an outcome or that this particular person had a 48.8% chance in some repeated personal trial.

The example's interpretation relies on the stated model being a reasonable representation of assignment across the relevant profiles. In particular, age and severity must be measured before treatment, and the specified functional forms must be plausible. The calculation alone cannot verify these conditions.

More Than Two Exposure Levels

When an exposure has \(K\) mutually exclusive levels, one can model the probability of each level. A common approach is multinomial logistic regression. Choose one level as the reference, here level 0, and specify a linear predictor \(\eta_j(X)\) for each other level \(j=1,\ldots,K-1\). The model is

P(A=j\mid X)= \frac{\exp\{\eta_j(X)\}} {1+\sum_{r=1}^{K-1}\exp\{\eta_r(X)\}}, \quad j=1,\ldots,K-1, \qquad P(A=0\mid X)= \frac{1}{1+\sum_{r=1}^{K-1}\exp\{\eta_r(X)\}}.

These probabilities are nonnegative and sum to one. To see why, write the denominator as \(D=1+\sum_{r=1}^{K-1}\exp\{\eta_r(X)\}\). Summing the probability for level 0 and all other levels gives

\frac{1}{D}+\sum_{j=1}^{K-1}\frac{\exp\{\eta_j(X)\}}{D} =\frac{1+\sum_{j=1}^{K-1}\exp\{\eta_j(X)\}}{D} =\frac{D}{D}=1.

The reference category is a modeling convention; changing it changes the coefficient representation, not the underlying fitted probabilities when the same model is expressed equivalently. This model assumes the specified log odds of each non-reference level versus the reference level are adequately represented by the chosen predictors and functional forms. As with a binary model, the exposure categories should be mutually exclusive and exhaustive for the analysis, and the covariates should be measured before exposure.

Interpretation: For a fixed covariate profile \(X\), \(\hat e_j(X)\) estimates the probability of receiving exposure level \(j\), conditional on the model's measured covariates. For each profile, the estimated probabilities across all exposure levels sum to one. A score is specific to an exposure level; in a multi-level setting it is a vector of probabilities, not a single probability that summarizes every comparison.

Worked Example: Three Exposure Levels

Consider hypothetical data in which an exposure has three levels: no program (\(A=0\)), standard program (\(A=1\)), and intensive program (\(A=2\)). For one hypothetical covariate profile, suppose the fitted multinomial model gives \(\hat\eta_1=1.5\) and \(\hat\eta_2=0.9\), relative to no program. The denominator is

D=1+\exp(1.5)+\exp(0.9).

Using unrounded exponential values in the divisions, the estimated scores are

\hat e_0(X)=\frac{1}{1+\exp(1.5)+\exp(0.9)}\approx0.1259,

\hat e_1(X)=\frac{\exp(1.5)}{1+\exp(1.5)+\exp(0.9)}\approx0.5644, \qquad \hat e_2(X)=\frac{\exp(0.9)}{1+\exp(1.5)+\exp(0.9)}\approx0.3097.

The sum is approximately \(0.1259+0.5644+0.3097=1.0000\). For this measured profile, the model assigns the largest estimated probability to the standard program. The scores remain conditional probabilities from a fitted model; they do not by themselves establish that the program levels are causally comparable.

The multinomial model's log-odds interpretation can also be checked directly. For level 1 versus the reference level,

\frac{P(A=1\mid X)}{P(A=0\mid X)} = \frac{\exp\{\eta_1(X)\}/D}{1/D} =\exp\{\eta_1(X)\}.

Thus \(\eta_1(X)\) is the log odds of level 1 versus level 0 at profile \(X\). The same reasoning applies to each non-reference level.

Choosing a Useful Specification

The covariates and their forms determine what the scores can represent. A simple model may be too restrictive. If age has a curved relationship with assignment, a linear age term can misstate probabilities at some ages. If the relationship between severity and assignment differs by age, a model without an age-by-severity interaction cannot represent that difference. Candidate specifications should be informed by subject-matter knowledge and the causal structure, rather than selected only because they predict exposure accurately.

Include measured pre-exposure covariates that are appropriate for controlling confounding, as informed by the causal question and adjustment-set reasoning. Do not treat every available variable as automatically appropriate: a post-exposure variable may be affected by exposure and can distort the intended adjustment. Nor should a variable be excluded solely because its association with exposure is not statistically significant in the sample. The purpose is not simply to identify predictors of treatment choice; it is to represent assignment conditional on the covariates needed for the causal comparison.

Worked Example: Why Functional Form Matters

Suppose a hypothetical binary treatment model includes baseline age \(x\), measured in decades and centered at 5 decades. Two candidate models are considered for a hypothetical profile with \(x=7\), meaning age 70. Model A uses only a linear term, while Model B also includes a quadratic term:

\hat\eta_A=-1+0.20x, \qquad \hat\eta_B=-1+0.20x-0.10x^2.

For Model A, the linear predictor is \(-1+0.20(7)=0.40\), giving

\hat e_A(X)=\frac{\exp(0.40)}{1+\exp(0.40)} =\frac{1.4918}{2.4918}\approx0.5988.

For Model B, the predictor is \(-1+0.20(7)-0.10(7^2)=-4.50\), giving

\hat e_B(X)=\frac{\exp(-4.50)}{1+\exp(-4.50)} =\frac{0.01111}{1.01111}\approx0.0110.

The large difference is a consequence of the hypothetical coefficients, not evidence that the quadratic model is preferable. It illustrates that a specification choice can substantially alter estimated assignment probabilities. Whether curvature is plausible should be considered using the assignment context and the observed data; a fitted value should not be treated as trustworthy merely because a more complex term can be added.

In practice, the probability model should also be checked for sensible predictions in the covariate profiles relevant to the target population. Very small or very large estimated probabilities may flag limited overlap or model problems, but they do not identify the cause on their own. Detailed overlap and balance assessments are separate steps in a propensity-score analysis.

What a Propensity Score Does—and Does Not—Establish

A well-specified assignment model estimates the conditional distribution of exposure given the covariates it includes. Under the causal identification assumptions, including conditional exchangeability given an adequate set of measured pre-exposure covariates and positivity, propensity scores can support causal comparisons. These assumptions concern the study and its measured information; fitting a logistic or multinomial model cannot prove them. In particular, a propensity score cannot adjust for an important confounder that was not measured or omitted from the model.

Propensity scores also do not estimate the outcome response. Their coefficients describe modeled exposure assignment, not the effect of exposure on the outcome. The estimated scores may later be used in methods that aim to create comparable groups, but the scores alone do not show that comparability has been achieved. That requires examining the adjusted data and the covariate balance relevant to the intended comparison.

Common pitfall: “The assignment model has high predictive accuracy, so confounding has been removed.” This is not justified. Accuracy asks whether the model predicts observed exposure labels; causal adjustment requires appropriate pre-exposure covariates, defensible specification, adequate overlap, and the relevant identification assumptions. A model can predict exposure well while omitting a confounder, and a model need not maximize exposure prediction to be useful for a causal analysis.

A Practical Modeling Sequence

Before interpreting scores, make the assignment model and its role explicit. The following sequence keeps the probability calculation distinct from the causal conclusions that may eventually depend on it.

Define the exposure levels.
State which categories are being compared, how they are coded, and which category is the reference if using multinomial logistic regression.

Specify pre-exposure covariates.
Use the causal question and measured covariates to decide what belongs in the assignment model; consider plausible nonlinearities and interactions.

Fit the appropriate probability model.
Use logistic regression for a binary exposure or a suitable multi-category model for mutually exclusive exposure levels, and account for the study's data structure.

Calculate and inspect estimated probabilities.
Confirm that probabilities are in the interval from 0 to 1 and, for multiple levels, sum to 1 for each profile. Interpret them as conditional assignment probabilities under the fitted specification.

The central discipline is to keep three statements separate: the model estimates exposure probabilities from measured covariates; the estimated scores depend on the model specification; and causal interpretation requires assumptions beyond successful model fitting. Stating all three prevents the propensity score from being mistaken for a treatment effect or for proof that observational groups are exchangeable.

Check Your Understanding

Try these questions without looking back.

In a binary exposure study, what does \(P(A=1\mid X)\) represent, and what is its relationship to \(P(A=0\mid X)\)?
A fitted multinomial model gives probabilities 0.20, 0.50, and 0.25 for three exposure levels at one profile. What check should be made before interpreting them?
Why does a high rate of correct exposure classification not establish that confounding has been removed?
Give one reason to consider a nonlinear age term or an interaction in an exposure-assignment model.
Can an estimated propensity score account for an unmeasured confounder? Explain briefly.